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Motion of level sets by mean curvature. I. (English) Zbl 0726.53029
Let g: $${\mathbb{R}}^ n\to {\mathbb{R}}$$ be a continuous function which defines an initial hypersurface $$\Gamma_ 0$$ by $$\Gamma_ 0=\{x\in {\mathbb{R}}^ n;\quad g(x)=0\}.$$ Consider the parabolic PDE $$u_ t=(\delta_{ij}- u_{x_ i}u_{x_ j}/| Du|^ 2)u_{x_ ix_ j}\text{ in } {\mathbb{R}}^ n\times [0,\infty),\quad u=g\text{ on } {\mathbb{R}}^ n\times \{t=0\}.$$One says that each level set of u evolves according to its mean curvature. S. Osher and J. A. Sethian [see J. Comput. Phys. 79, No.1, 12-49 (1988; Zbl 0659.65132)] have introduced various techniques to study the above equation numerically.
In the paper under review, the authors introduce the definition of a weak solution of this equation. Their definition agrees with the classical motion by mean curvature. The existence and uniqueness of a weak solution are proved. Define $$\Gamma_ t=\{x\in {\mathbb{R}}^ n;\quad u(x,t)=0\}.\{\Gamma_ t\}_{t>0}$$ is said to be the generalized evolution by mean curvature of the original compact set $$\Gamma_ 0$$. Geometric properties of generalized evolution by mean curvature are obtained. The last section contains examples of pathologies and conjectures.

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58J35 Heat and other parabolic equation methods for PDEs on manifolds 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations
##### Keywords:
mean curvature; weak solution; evolution
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